The key here is plugging in, rather than relying on absolute mathematical concepts, which may contain end cases that you may not think about considering. Here's how I would approach the question:

When reading the question stem, the possible answers are "yes, |x| equals y-z" or "No, |x| is not equal y-z". In DS yes/No questions, your initial approach is to try and show that the statement(s) allow for both a yes and a no, thus proving the statement insufficient. Only if you cannot reach opposite results should you then take a step back and see why the statement(s) is actually sufficient.

The absolute value in the question stem is a clue to plugging in positive and negative values for x, since |x| is always positive, but x itself may be negative: the absolute value of -5 is the same as absolute value of 5 - both are equal to 5.

So:
Stat. (1): x+y=z, or x=z-y.
If z=3 and y=2, then x=3-2=1, and then |1| is NOT equal to the question stem's y-z=2-1=-3. that's a no. Can we find a yes?
Try to plug in values of z and y that will yield a negative x:
if z=2 and y=5, then x=2-5=-3. In this case, |-3|=3 IS equal to the question stem's y-z=5-2=3, so the answer is "yes".
Since you have both a "yes" and "no" possible answers to the question, the answer is maybe: stat. (1)->Maybe-> IS.

Stat. (2): x <0 alone is meaningless.

combined: Stat. (2) effectively eliminates the "no" scenario we tried earlier (and others like it): x has to be negative, so z-y has to be negative. The absolute value of x will then be the opposite, positive value of the negative expression z-y, which is equal to -(z-y) = y-z.